Re: Ludwig Boltzmann, entropy

Tim Tyler <tim@xxxxxxxxxxx> wrote:
>Paul J Gans <gans@xxxxxxxxx> wrote or quoted:
>> Tim Tyler <tim@xxxxxxxxxxx> wrote:
>> >Paul J Gans <gans@xxxxxxxxx> wrote or quoted:
>> >> Tim Tyler <tim@xxxxxxxxxxx> wrote:

>> >> >Thermodynamic entropies may well prove to be totally inaccurate
>> >> >as measures of the real entropy of systems. They could be
>> >> >out by orders of magnitudes - not just by a few significant
>> >> >figures. For example, they completely ignore the possibility
>> >> >of there being significant entropy within atoms.
>>
>> >[...]
>>
>> >> This is both right and wrong. If we heat a system hot enough
>> >> we *can* excite nuclear modes. There are nuclear energy levels.
>> >>
>> >> However, the entropy of the system is not affected *until*
>> >> we start to excite such modes.
>>
>> >To go back to the start for a moment, in thermodynamics, a microstate is
>> >usually defined as a state in which the motions of the individual
>> >particles in the system are completely specified. How much information
>> >does it take to completely specify a particle's velocity? Nobody knows.
>>
>> What *are* you talking about. First, that's not the definition
>> of a microstate these days.


>Maybe we have a different definition of microstate. That might
>explain our differences. I've stated mine - what's yours.

A microstate is basically a quantum state.

>> Second, it takes three numbers to define a particle's velocity in three
>> dimensions. So the "nobody knows" is wrong.

>"Three numbers" does not specify a quanity of information -
>when the numbers are decimal point representations of unknown
>precision. Since your premise was wrong, so is the conclusion.

You misread me. It takes THREE numbers. If your point is
that we cannot specify any of the components of the velocity
vector to infinite precision, you are right. We cannot.

It also makes no difference at all because quantum mechanics
tells us that if we *could* specify a velocity that precisely
we would have no idea where the particle was in the universe.

>> If you were right we'd not be able to do Newtonian mechanics,
>> and we've been doing that since the 17th century. [...]

>An unjustified and incorrect assertion.

Why? We can calculate the orbit of Mars quite
satisfactorily knowing three velocity components for
each body in the solar system. And we don't need
infinite precision.


>> >There are some guesses based on the hypothesis that the spacetime
>> >becomes discrete down at around the Planck length - but these are
>> >just speculation at this stage.
>>
>> Now you are confusing classical and a quantal descriptions
>> of a system. The discrete or lack of discrete nature of
>> lengths is quite beside the point.

>You need to be able do distinguish between microstates if you
>are going to count them. Microstates *must* be discrete.
>If they are not, there is no way to count them.

That's a bit of a problem. One ought to study the philosophy
of quantum mechanics before saying things like that.


>> >So: nobody knows how many different possible microstates there are
>> >available to any physical system. We have no idea how many possible
>> >velocities there are for any particle. We don't even know if the
>> >figure is finite or not.
>>
>> What *are* you talking about. For a number of systems we
>> can calculate the number of microstates from basic theory.
>> For other systems, if we can measure the entropy (and we
>> almost always can) we can then get the number of microstates
>> from that.
>>
>> So we have two ways -- theoretical and experimental. What
>> don't you like about that?

>It's wrong.

What's wrong? I calculated the number of microstates in
a room temperature diamond for you in a previous post.
Did I make a math mistake?

>> In three-dimensional space there are exactly three components
>> of velocity needed to specify the "velocity". Three is a
>> finite number.

>As I mentioned, "three" is not a measure of the information involved.

Nor did I say it was. I said we needed THREE numbers.


>> If you are worried about the number of digits in the velocity
>> components, that's not really important.

>It's critical.

Not at all. The uncertainty principle guarantees that
the conjugate errors in position and momentum (to be
technically correct before I get nit-picked) are bounded
by Planck's constant. If you specify the error in one
quantity, the error in the other cannot be reduced beyond
a certain limit no matter how hard you try.

As a result there is no need or use in specifying positions
and momenta beyond a certain amount.


>> In both classical and quantal statistical mechanics we *must* smear out
>> the velocities a bit. The joint specification of a component
>> of a velocity times the component of position has an
>> uncertainty of the order of Planck's constant. Thus there
>> is no need to specify these to more than that precision.

>So: you are assuming that systems have velocities that
>are discrete on the level of Planck's constant - and that
>two systems that are indistinguishable on that scale are
>identical - and can be described as being in the same state.

No. You have not stated my position correctly. I'm dealing
in particles and have been. Two particles that have the
same positions and momenta to within the limits set by
the uncertainty principle are in fact totally indistinguishable
in a very profound sense. You can't paint a number on the
side of a particle. You can only identify them by their
positions and momenta, and that to only within the limit
set by the uncertainty principle.

>That is an important difference between our positions. I am not
>making that assumption. The universe may be discrete on that level.
>It may be discrete on a higher level than that. Or it may be
>discrete on a lower level than that. Or it may not be discrete
>at all. Nobody knows which of those positions is correct.

That's silly. First, do NOT make statements about the
universe. Talk about systems or better yet particles.
We know that energies in spatially bounded systems are
not continuous. So "it may not be discrete at all" is
out. We also know that it is not discrete at a level
higher than quantum mechanics or we'd have observed it.
So those are out.

Might there be further structure below quantum mechanics?
That's been thought about for some decades now. It is
called "hidden variable theory". Some fairly stringent
restrictions have been placed on hidden variables to
the point where most physicists believe that they
do not exist. Thus we are stuck with quantum mechanics.

>The other difference between our positions seems to be that
>you are using a definition of entropy that *only* depends on
>particle positions and momenta. However, those may not be
>the only attributes particles can have which can vary
>(indeed we know that they are not).

Oh? And what might those attributes be?


>Since for one thing, we don't actually know what the laws
>of physics are, there may be mountains of other relevant
>information present in physical systems, which is not
>included by measuring positions and momenta.

>I am not assuming this information does not exist, or is
>of negligible size. We don't know the magnitude of this
>information at all at this stage. There may be ways of
>using this sort of information to do work.

Is this an argument like "we don't know the velocity to
infinite precision"? You make it sound like we know
nothing. This is far from the truth. We know a good
bit about the laws of physics. Some of what we know
is likely to be only a good approximation. Some small
bits might even be wrong.

But we can assert that when it comes to the entropy
we understand the situation fairly well.

Let me remind you that thermodynamics existed BEFORE
quantum mechanics and it was not changed by the
development of quantum mechanics. This is because
it is an axiomatic system like plane geometry. It
can only be wrong if the axioms are wrong.

Statistical mechanics has one foot resting on thermodynamics
and one foot resting on quantum mechanics.

Quantum mechanics is not quite as firm as thermodynamics,
but it is really pretty damn firm. We've had almost a
century of quanta now and with the possible exception
of gravity, no real problems with it.

As a result statistical mechanics is on a rather good
foundation. It's results are mathematically correct
and agee with experiment.


>We don't know how many particles there are in any region
>of space either. How many neutrinos are there in the
>diamond you mentioned. How many gravitons? Each particle
>contributes to the entropy of the system and the number of
>possible states it can take up - yet nobody has a clear
>idea of how many particles there are in any region of
>spacetime.

This is wrong. We don't know about them because they
*don't* contribute to the thermodynamics of the system
at all.

You have got to come to grips with this. Things like
energy and entropy depends on the ability of a system
to move from one energy state to another. Neutrinos,
for example, are so weakly coupled to other forms
of matter that we can't do *anything* to them short
of using a mountain-sized neutrino detector.

In other words, they just don't count.

>These sorts of consideration make a *big* difference
>to the number of different states physical systems can occupy.

Nope. They make no difference to the number of
quantum states that matter.

Let me give you an example that will confuse you
horribly. Pick a number, any number, no matter
how large.

There is a non-zero probability that the next
molecule to whiz by your nose has that energy,
even if it is more than the total mass energy
of the sun.

Post the number and I'll be happy to calculate
the probability for you. Or you could get any
of a large number of other folks here to do it.

It isn't zero.

Are you going to not be able to sleep for worrying
about this?

----- Paul J. Gans

.

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